Parabolic subgroups in FC-type Artin groups

Morris-Wright, Rose

doi:10.1016/j.jpaa.2020.106468

Cited by 24 publications

(31 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…We already point out that part (i) can be proven using the same argument as given in [24,Lemma 5.2]; actually, we have the following lemma. Lemma 1.4.…”

Section: Introductionmentioning

confidence: 80%

“…In [13], Cumplido, Gebhardt, González-Meneses and Wiest explain a one-to-one correspondence between isotopy classes of simple closed curves in D n+1 and proper irreducible parabolic subgroups of A An which allows them to translate the definition of the curve graph CG(D n+1 ) in purely algebraic terms. This definition can then be generalized to any Artin-Tits group as follows: Definition 1.1 [13] [24,Definition 4.1]. Let A Γ be an Artin-Tits group.…”

Section: Introductionmentioning

confidence: 99%

“…Note that

{scriptC}_{p a r a b} false(normalΓ false)

is empty if

normalΓ

consists of a single vertex (

A_{Γ}

is cyclic). If

A_{Γ}

is dihedral, then by [24, Lemma 5.2, Theorem 5.3],

{scriptC}_{p a r a b} false(normalΓ false)

is not connected and has infinite diameter, unless

normalΓ

consists of two vertices with no edge, in which case

A_{normalΓ} ≅ {double-struckZ}^{2}

and

{scriptC}_{p a r a b} false(normalΓ false)

consists of two vertices and a single edge between them. Also, if

normalΓ

is not connected,

{scriptC}_{p a r a b} false(normalΓ false)

is easily shown to have diameter 2.…”

Section: Introductionmentioning

confidence: 99%

“…Most of the known properties of the graph

{scriptC}_{p a r a b} false(normalΓ false)

are gathered in [11] and [24]. For example, if

A_{Γ}

is irreducible and of spherical type (

normalΓ

with at least three vertices), then

{scriptC}_{p a r a b} false(normalΓ false)

is connected and has infinite diameter ([24, Lemma 5.2] and [11, Corollary 4.13]).…”

Section: Introductionmentioning

confidence: 99%

“…Most of the known properties of the graph

{scriptC}_{p a r a b} false(normalΓ false)

are gathered in [11] and [24]. For example, if

A_{Γ}

is irreducible and of spherical type (

normalΓ

with at least three vertices), then

{scriptC}_{p a r a b} false(normalΓ false)

is connected and has infinite diameter ([24, Lemma 5.2] and [11, Corollary 4.13]). Finally, when

Γ = A_{n}

{scriptC}_{p a r a b} false(A_{n} false)

is isomorphic to the curve graph

CG ({double-struckD}_{n + 1})

and it is hyperbolic in virtue of Masur–Minsky's theorem.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Curve graphs for Artin–Tits groups of type B, A∼ and C∼ are hyperbolic

Calvez

Cruz

2021

Trans London Maths Soc

View full text Add to dashboard Cite

The graph of irreducible parabolic subgroups is a combinatorial object associated to an Artin-Tits group A defined so as to coincide with the curve graph of the (n + 1)-times punctured disk when A is Artin's braid group on (n + 1) strands. In this case, it is a hyperbolic graph, by the celebrated Masur-Minsky's theorem. Hyperbolicity of the graph of irreducible parabolic subgroups for more general Artin-Tits groups is an important open question. In this paper, we give a partial affirmative answer.For n 3, we show that the graph of irreducible parabolic subgroups associated to the Artin-Tits group of spherical type Bn is also isomorphic to the curve graph of the (n + 1)-times punctured disk; hence, it is hyperbolic.For n 2, we show that the graphs of irreducible parabolic subgroups associated to the Artin-Tits groups of euclidean type An and Cn are isomorphic to some subgraphs of the curve graph of the (n + 2)-times punctured disk which are not quasi-isometrically embedded. We prove nonetheless that these graphs are hyperbolic.

show abstract

“…We already point out that part (i) can be proven using the same argument as given in [24,Lemma 5.2]; actually, we have the following lemma. Lemma 1.4.…”

Section: Introductionmentioning

confidence: 80%

Section: Introductionmentioning

confidence: 99%

“…Note that

{scriptC}_{p a r a b} false(normalΓ false)

is empty if

normalΓ

consists of a single vertex (

A_{Γ}

is cyclic). If

A_{Γ}

is dihedral, then by [24, Lemma 5.2, Theorem 5.3],

{scriptC}_{p a r a b} false(normalΓ false)

is not connected and has infinite diameter, unless

normalΓ

consists of two vertices with no edge, in which case

A_{normalΓ} ≅ {double-struckZ}^{2}

and

{scriptC}_{p a r a b} false(normalΓ false)

consists of two vertices and a single edge between them. Also, if

normalΓ

is not connected,

{scriptC}_{p a r a b} false(normalΓ false)

is easily shown to have diameter 2.…”

Section: Introductionmentioning

confidence: 99%

“…Most of the known properties of the graph

{scriptC}_{p a r a b} false(normalΓ false)

are gathered in [11] and [24]. For example, if

A_{Γ}

is irreducible and of spherical type (

normalΓ

with at least three vertices), then

{scriptC}_{p a r a b} false(normalΓ false)

is connected and has infinite diameter ([24, Lemma 5.2] and [11, Corollary 4.13]).…”

Section: Introductionmentioning

confidence: 99%

“…Most of the known properties of the graph

{scriptC}_{p a r a b} false(normalΓ false)

are gathered in [11] and [24]. For example, if

A_{Γ}

is irreducible and of spherical type (

normalΓ

with at least three vertices), then

{scriptC}_{p a r a b} false(normalΓ false)

is connected and has infinite diameter ([24, Lemma 5.2] and [11, Corollary 4.13]). Finally, when

Γ = A_{n}

{scriptC}_{p a r a b} false(A_{n} false)

is isomorphic to the curve graph

CG ({double-struckD}_{n + 1})

and it is hyperbolic in virtue of Masur–Minsky's theorem.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Curve graphs for Artin–Tits groups of type B, A∼ and C∼ are hyperbolic

Calvez

Cruz

2021

Trans London Maths Soc

View full text Add to dashboard Cite

show abstract

The Conjugacy Stability Problem for Parabolic Subgroups in Artin Groups

Cumplido

2022

Mediterr. J. Math.

View full text Add to dashboard Cite

Given an Artin group A and a parabolic subgroup P, we study if every two elements of P that are conjugate in A, are also conjugate in P. We provide an algorithm to solve this decision problem if A satisfies three properties that are conjectured to be true for every Artin group. This allows to solve the problem for new families of Artin groups. We also partially solve the problem if A has FC-type, and we totally solve it if A is isomorphic to a free product of Artin groups of spherical type. In particular, we show that in this latter case, every element of A is contained in a unique minimal (by inclusion) parabolic subgroup.

show abstract

Extra-large type Artin groups are hierarchically hyperbolic

2022

View full text Add to dashboard Cite

We show that Artin groups of extra-large type, and more generally Artin groups of large and hyperbolic type, are hierarchically hyperbolic. This implies in particular that these groups have finite asymptotic dimension and uniform exponential growth. We prove these results by using a combinatorial approach to hierarchical hyperbolicity, via the action of these groups on a new complex that is quasi-isometric both to the coned-off Deligne complex introduced by Martin–Przytycki and to a generalisation due to Morris-Wright of the graph of irreducible parabolic subgroups of finite type introduced by Cumplido–Gebhardt–González-Meneses–Wiest.

show abstract

Parabolic subgroups in FC-type Artin groups

Cited by 24 publications

References 17 publications

Curve graphs for Artin–Tits groups of type B, A∼ and C∼ are hyperbolic

Curve graphs for Artin–Tits groups of type B, A∼ and C∼ are hyperbolic

The Conjugacy Stability Problem for Parabolic Subgroups in Artin Groups

Extra-large type Artin groups are hierarchically hyperbolic

Contact Info

Product

Resources

About